I've always felt that the way the question is put is an incomplete sentence. "Λυπάμαι Ελένη, αλλά ..." "I regret it Helen but ..." I've always felt that when you use Λυπάμαι there is an implication for more or the sentence is not complete. "Sorry Helen" translates to "Συγνώμη Ελένη". A better use of Λυπάμαι might be "Το λυπάμαι Ελένη" " I regret it Helen"

Because this is the vocative case which does not take an article. The vocative is used to call someone, so when you address someone saying 'hey you', that 'you' is in vocative.
Here, Eleni is not sorry herself (that would be the nominative η Ελένη), you're not sorry for her (accusative, την Ελένη), it's not about something that is Eleni's (της Ελένης), so we're left with the vocative (_ Ελένη!).